Question

Solve the equation.
4 cos² x = 1

Answer 1

cos^2 x=1/4

cosx =1/2

cosx =cos 60°

x=2n pi + or-60°

Answer 2

Solution of equation:

$4 {cos}^{2} x = 1 \\ \\ {cos}^{2}x = \frac{1}{4} \\ \\ \cos(x) = \sqrt{ \frac{1}{4} } \\ \\ \cos(x) = ± \frac{1}{2}$

So there are two values of x can be calculated

$\cos(x) = \frac{1}{2} \\ \\ x = {cos}^{ - 1} ( \frac{1}{2} ) \\ \\ x = {cos}^{ - 1} (cos \: ( \frac{\pi}{3} ) \\ \\ = \frac{\pi}{3} \: \: belongs \: to \: [0,\pi]$

$\cos(x) = - \frac{1}{2} \\ \\ x = {cos}^{ - 1} ( - \frac{1}{2} ) \\ \\ x = {cos}^{ - 1} ( - cos \: ( \frac{\pi}{3} ) \\ \\ = {cos}^{ - 1} (cos(\pi - \frac{\pi}{3} ) \\ \\ = {cos}^{ - 1}(cos( \frac{2\pi}{3})) \\ \\= \frac{2\pi}{3} \: \: belongs \: to \: [0,\pi]$

Thus two values of x are π/3 and 2π/3.